234 C. Argyris et al. which are evaluated analytically for the given form of the likelihood and prior distributions. Then the posterior variance of θ is given by (26.7) as: Var(θ| dx) =E[θ 2| dx]−E[θ| dx] 2 (26.21) and the final objective function is given by the integral over the data: Ux = +∞ −∞ p( dx)Var(θ| dx)dy (26.22) Carrying out all the integrations analytically for the three cases of error standard deviation results in the following forms for the objective function respectively: U 1 x = σ 2 e σ 2 p σ 2 e +σ 2 px 2 (26.23) U 2 x = σ 2 e σ 2 p σ 2 e +σ 2 p (26.24) U 3 x = σ 2 e σ 2 p σ 2 e +σ 2 px 4 (26.25) Next we interpret the resulting objective functions. In the absence of measurement error e, any location x would be equally good to find the value of θ simply by solving the model equation for θ: θ =y/x. This intuitive notion is reflected in all the objective functions by reducing to zero for σ e =0 independently of the value of x. A zero objective function means zero expected posterior variance for θ, which means that we learn the value of θ exactly with no uncertainty at all. Also note that the objective functions reduce to zero for σp = 0 independently of the value of x again. This is because a zero prior uncertainty implies that we already know exactly the value of θ and no additional data can change that; that is, the posterior is always dominated by the prior. However, for non-zero values of σ e andσp there is dependence on the measured locationx in objective functions U 1 x and U 3 x which correspond to the constant and inversely proportional error cases respectively. Specifically, we see that the objective function decreases (posterior uncertainty inθ decreases) as x increases. This is because the farther away we measure the less important is the measurement error compared to the actual model value. In the constant error case, the error remains the same as the model output is increased when we increase x and therefore the signal-to-noise ratio gets larger. Therefore we have more accurate data dx which is contaminated with less noise, which in turn leads to greater accuracy in the identified valueof θ. The same principle holds in the inversely proportional error case but even stronger since the error does not remain constant, but it actually decreases with x, and this results in an even faster reduction of the objective function as x increases (x 4 compared tox 2). Finally note that for the trivial case of x =0, U 1 x andU 3 x reduce to the prior varianceσ 2 p. This is because at x =0 the model output is zero and we only measure noise which has no valuable information about θ, and therefore the prior uncertainty remains unchanged. In the proportional error case we see that U 2 x does not depend on x at all. This is due to the fact that the signal-to-noise ratio remains the same regardless of x, since the noise is always proportional tox. Small model outputs have small error, and large model outputs have large error, and therefore there is no preference to a specific x so the objective function is constant and independent of x. Finally note how the objective functions do not depend on the Gaussian prior mean mp but only on its variance σ 2 p since what matters is the posterior variance and not the mean. 26.5.1 Comparison with Monte Carlo Integration Next we solve the same problem (constant error case) using Monte Carlo integration for an increasing number of samples in order to check the convergence. The exact analytical solution U 1 x is known in this case and is also shown for comparison purposes. The values of σ e, σp and x which were used are shown in the titles of Fig. 26.2.
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